Low-complexity Minimum Mean Square Error Research Articles

Low-Complexity BFGS-Based Soft-Output MMSE Detector for Massive MIMO Uplink

For the massive multiple-input multiple-output (MIMO) uplink, the linear minimum mean square error (MMSE) detector is near-optimal but involves undesirable matrix inversion. In this paper, we propose a low-complexity soft-output detector based on the simplified Broyden–Fletcher–Goldfarb–Shanno method to realize the matrix-inversion-free MMSE detection iteratively. To accelerate convergence with minimal computational overhead, an appropriate initial solution is presented leveraging the channel-hardening property of massive MIMO. Moreover, we employ a low-complexity approximated approach to calculating the log-likelihood ratios with negligible performance losses. Simulation results finally verify that the proposed detector can achieve the near-MMSE performance with a few iterations and outperforms the recently reported linear detectors in terms of lower complexity and faster convergence for the realistic massive MIMO systems.

Low-Complexity MMSE Receiver Design for Massive MIMO OTFS Systems

We design a low-complexity minimum mean square error (MMSE) receiver for a practical rectangular pulse-shaped orthogonal time frequency space (OTFS) massive multiple-input multiple-output (mMIMO) system. The proposed receiver is designed by exploiting the OTFS channel sparsity, and the structure of matrices involved in the MMSE receiver. We show that the sparse OTFS channel matrix is multi-banded, and has a high bandwidth. This is because its non-zero elements, which occur in multiple bands, are scattered throughout the matrix. We reduce its bandwidth by using the Reverse Cuthill-Mckee algorithm. The proposed MMSE receiver does not use any approximation, and thus has the same bit error rate (BER) as that of its conventional counterpart. We show that the proposed receiver, without any BER loss, achieves significantly lower complexity than the existing state-of-the-art receivers.

Low-complexity and high-performance receive beamforming for secure directional modulation networks against an eavesdropping-enabled full-duplex attacker

In this paper, we present a novel scenario for directional modulation (DM) networks with a full-duplex (FD) malicious attacker (Mallory), where Mallory can eavesdrop the confidential message from Alice to Bob and simultaneously interfere Bob by sending a jamming signal. Considering that the jamming plus noise at Bob is colored, an enhanced receive beamforming (RBF), whitening-filter-based maximum ratio combining (MRC) (WFMRC), is proposed. Subsequently, two RBFs of maximizing the secrecy rate (Max-SR) and minimum mean square error (MMSE) are presented to show the same performance as WFMRC. To reduce the computational complexity of conventional MMSE, a low-complexity MMSE is also proposed. Eventually, to completely remove the jamming signal from Mallory and transform the residual interference plus noise to a white one, a new RBF, null-space projection (NSP) based maximizing WF receive power, called NSP-based Max-WFRP, is also proposed. From simulation results, we find that the proposed Max-SR, WFMRC, and low-complexity MMSE have the same SR performance as conventional MMSE, and achieve the best performance while the proposed NSP-based Max-WFRP performs better than MRC in the medium and high signal-to-noise ratio regions. Due to its low-complexity,the proposed low-complexity MMSE is very attractive. More important, the proposed methods are robust to the change in malicious jamming power compared to conventional MRC.

Bayesian Learning Aided Sparse Channel Estimation for Orthogonal Time Frequency Space Modulated Systems

A novel sparse channel state information (CSI) estimation scheme is proposed for orthogonal time frequency space (OTFS) modulated systems, in which the pilots are directly transmitted over the time-frequency (TF)-domain grid for estimating the delay-Doppler (DD)-domain CSI. The proposed CSI estimation model leads to a reduction in the pilot overhead as well as the training duration required. Furthermore, it does not require a DD-domain guard interval between the pilot and data symbols, hence increasing the bandwidth efficiency. A novel Bayesian learning (BL) framework is proposed for CSI acquisition, which exploits the DD-domain sparsity for improving the estimation accuracy in comparison to the conventional minimum mean squared error (MMSE)-based scheme. A low-complexity linear MMSE detector is used in the subsequent data detection phase. Our simulation results demonstrate the performance improvement of the proposed BL-based scheme over the conventional MMSE-based scheme as well as over other existing sparse estimation schemes.

Max-Min Fair Precoder Design and Power Allocation for MU-MIMO NOMA

In this correspondence, transmit precoders and their power allocation coefficients are designed jointly for a downlink non-orthogonal multiple access (NOMA) wireless communication system. The objective is to provide max-min fairness (MMF) among the strongest users in each group, while maintaining minimum target rates for all the other users. The proposed solutions are respectively based on semi-definite relaxation (SDR) and successive convex approximation (SCA) and on the minimum mean square error (MMSE) approaches. For the latter approach, a simplification is also suggested to lower complexity. It is shown that while rate-splitting (RS) has the best MMF rates, the low-complexity MMSE approach has the least complexity. Moreover, the SDR/SCA approach offers an excellent tradeoff between the two.

Low-Complexity MIMO MMSE Receiver with Performance Enhancement via Coordinate Interleaving

Multiple-input multiple-output (MIMO) Minimum mean-square error (MMSE) receivers are widely adopted in the latest communication standards and reducing the complexity of these receivers while preserving the error performance is highly desirable. In this work, we study the error performance and implementation complexity of MIMO MMSE receivers when combined with a coordinate interleaved signal space diversity (SSD) technique. Contrary to the well-known trade-off between the error performance and implementation complexity, the proposed system leads to a considerably simplified MIMO MMSE receiver with significant performance gains when compared to the original MIMO MMSE receiver. Unlike the standard MIMO MMSE receiver, the proposed coordinate interleaved technique induces a block diagonal transmit correlation matrix providing both performance enhancement and complexity reduction. The results show that the error performance can be improved more than 10[Formula: see text]dB with up to 71% computational complexity reduction. The complexity comparison between the original and proposed approaches is also verified by means of field-programmable gate array (FPGA) implementation.

Low-Complexity Subspace MMSE Channel Estimation in Massive MU-MIMO System

Massive multi-user multiple-input multiple-output (massive MU-MIMO) technology is considered as a promising enabler to fulfill the rapid growth of traffic requirement for wireless mobile communications. The massive MU-MIMO system can achieve unlimited capacity when the base station (BS) has accurate channel state information (CSI). In time-division-duplex (TDD) mode, the BS estimates CSI by receiving pilot signals sent from user terminals (UEs). However, because of using non-orthogonal pilots, pilot contamination happens to degrade the quality of the CSI estimation. To deal with pilot contamination problem, a low-complexity subspace minimum mean square error (MMSE) estimation method is proposed in this paper. Specifically, our approach operates the MMSE estimation in a low-dimensional subspace to avoid large matrix manipulation. Meanwhile, subspace projection helps to discriminate the desired signal and interfering signals in the power domain. Interference analysis shows the MMSE estimation can achieve interference-free estimation even in a low-dimensional subspace with a large number of BS antennas, and non-overlapping angles of arrival (AoAs) between desired and interfering UEs. Furthermore, thanks to the low-rank property of the channel covariance matrix in massive MU-MIMO systems, a two-step covariance matrix subspace projection method is proposed for further computational complexity reduction. The complexity analysis and simulation results indicate that our proposed approach has better channel estimation accuracy with lower complexity than the conventional MMSE estimation when the number of BS antennas is large.

Filtered Multitone Modulation Underwater Acoustic Communications Using Low-Complexity Channel-Estimation-Based MMSE Turbo Equalization

Filtered multitone (FMT) modulation divides the communication band into several subbands to shorten the span of symbols affected by multipath in underwater acoustic (UWA) communications. However, there is still intersymbol interference (ISI) in each subband of FMT modulation degrading communication performance. Therefore, ISI suppression techniques must be applied to FMT modulation UWA communications. The suppression performance of traditional adaptive equalization methods often exploited in FMT modulation UWA communications is limited when the effect of ISI spans tens of symbols or large constellation sizes are used. Turbo equalization consisting of adaptive equalization and channel decoding can improve equalization performance through information exchanging and iterative processes. To overcome the shortcoming of traditional minimum mean square error (MMSE) equalization and effectively suppress the ISI with relatively low computation complexity, an FMT modulation UWA communication using low-complexity channel-estimation-based (CE-based) MMSE turbo equalization is proposed in this paper. In the proposed method, turbo equalization is first exploited to suppress the ISI in FMT modulation UWA communications, and the equalizer coefficients of turbo equalization are adjusted using the low-complexity CE-based MMSE algorithm. The proposed method is analyzed in theory and verified by simulation analysis and real data collected in the experiment carried out in a pool with multipath propagation. The results demonstrate that the proposed method can achieve better communication performance with a higher bit rate than the FMT modulation UWA communication using traditional MMSE adaptive equalization.

Low-complexity MMSE detector based on refinement Jacobi method for massive MIMO uplink

Minimum mean square error (MMSE) linear detector can achieve the near-optimal bit error rate (BER) performance for uplink multi-user massive multiple input multiple output (MIMO) systems, but it involves matrix inversion with high complexity, especially when the number of users is high. In order to reduce the complexity, Jacobi iterative method has been applied for low-complexity MMSE detector without employing the computationally intensive matrix inversion. However, its convergence rate is low. In this paper, we propose a novel low-complexity MMSE detector based on the refinement of Jacobi iterative method. This refinement is given by the use of band matrix in order to accelerate the convergence rate, as well as guaranteeing the near-optimal BER performance. Analytical and simulation results show the efficiency of the proposed detector in comparison to the recently Jacobi-based detector.

Matrix Characterization for GFDM: Low Complexity MMSE Receivers and Optimal Filters

In this paper, a new matrix-based characterization of generalized-frequency-division-multiplexing (GFDM) transmitter matrices is proposed, as opposed to traditional vector-based characterization with prototype filters. The characterization facilitates deriving properties of GFDM (transmitter) matrices, including conditions for GFDM matrices being nonsingular and unitary, respectively. Using the new characterization, the necessary and sufficient conditions for the existence of a form of low-complexity implementation for a minimum mean square error (MMSE) receiver are derived. Such an implementation exists under multipath channels if the GFDM transmitter matrix is selected to be unitary. For cases where this implementation does not exist, a low-complexity suboptimal MMSE receiver is proposed, with its performance approximating that of an MMSE receiver. The new characterization also enables derivations of optimal prototype filters in terms of minimizing receiver mean square error (MSE). They are found to correspond to the use of unitary GFDM matrices under many scenarios. The use of such optimal filters in GFDM systems does not cause the problem of noise enhancement, thereby demonstrating the same MSE performance as orthogonal frequency division multiplexing. Moreover, we find that GFDM matrices with a size of power of two are verified to exist in the class of unitary GFDM matrices. Finally, while the out-of-band (OOB) radiation performance of systems using a unitary GFDM matrix is not optimal in general, it is shown that the OOB radiation can be satisfactorily low if parameters in the new characterization are carefully chosen.

Low Complexity ZF and MMSE Detectors for the Uplink MU-MIMO Systems With a Time-Varying Number of Active Users

The classical multiuser detection algorithms such as zero forcing (ZF) and minimum mean square error (MMSE) receivers are designed with the assumption that the number of active users is constant and known. When the number of the active users changes, the receiver may exhibit a serious performance loss if it does not react quickly to such variations. In this paper, we address the problem of reducing the complexity of the reevaluation of the popular ZF and MMSE detectors for multiuser multiple-input multiple-output (MU-MIMO) systems with a time-varying number of users in the channel. For each technique, we propose a detection approach with low complexity and without performance loss. The proposed algorithms avoid the direct computation of matrix inverses required by the ZF and MMSE detectors. Moreover, the performance losses, due to the use of the ZF and MMSE detectors intended for the scenario with a fixed number of active users, are evaluated with Monte Carlo simulation results.

Low-Complexity MMSE Precoding for Coordinated Multipoint With Per-Antenna Power Constraint

We propose a low-complexity minimum mean square error (MMSE) transmit filter design for the coordinated beamforming (CB) in the coordinated multipoint (CoMP) under the practical per-antenna power constraint (PAPC). The proposed design is based on the nonlinear Gauss-Seidel type algorithm in which the transmit filters for given receive filters are computed by iteratively updating the beamformer of each transmit antenna using simple closed-form expressions. The proposed approach can significantly reduce the overall complexity of the alternating optimization while preserving the optimality in the MSE sense.

Low-complexity SAGE-based MMSE channel estimation for CoMP multi-user systems

Coordinated multiple point (CoMP) transmission/reception has been investigated recently as a promising technology to increase the cell-edge user performance of long term evolution-advanced (LTE-A), and channel estimation is a crucial technology for CoMP systems. In this paper, we consider a reduced-complexity minimum mean square error (MMSE) channel estimator for CoMP systems. The estimator uses space alternating generalized expectation maximization (EM) (SAGE) algorithm to avoid the inverse operation of the joint MMSE estimator. In the proposed scheme, the base stations (BSs) in the CoMP system estimate the channels of all the coordinated users serially and iteratively. We derive the SAGE-based estimators and analyze complexity. Simulation results verify that the performance of the proposed algorithm is close to the joint MMSE estimation algorithm while reducing the complexity greatly.

Joint Time-Frequency Diversity for Single-Carrier Block Transmission in Frequency Selective Channels

In time-varying frequency selective channels, to obtain high-rate joint time-frequency diversity, linear dispersion coded orthogonal frequency division multiplexing (LDC-OFDM), has recently been proposed. Compared with OFDM systems, single-carrier systems may retain the advantages of lower PAPR and lower sensitivity to carrier frequency offset (CFO) effects, which motivates this paper to investigate how to achieve joint frequency and time diversity for high-rate single-carrier block transmission systems. Two systems are proposed: linear dispersion coded cyclic-prefix single-carrier modulation (LDC-CP-SCM) and linear dispersion coded zero-padded single-carrier modulation (LDC-ZP-SCM) across either multiple CP-SCM or ZP-SCM blocks, respectively. LDC-SCM may use a layered two-stage LDC decoding with lower complexity. This paper analyzes the diversity properties of LDC-CP-SCM, and provides a sufficient condition for LDC-CP-SCM to maximize all available joint frequency and time diversity gain and coding gain. This paper shows that LDC-ZP-SCM may be effectively equipped with low-complexity minimum mean-squared error (MMSE) equalizers. A lower complexity scheme, linear transformation coded SCM (LTC-SCM), is also proposed with good diversity performance.

ASIC Implementation of Soft-Input Soft-Output MIMO Detection Using MMSE Parallel Interference Cancellation

Multiple-input multiple-output (MIMO) technology is the key to meet the demands for data rate and link reliability of modern wireless communication systems, such as IEEE 802.11n or 3GPP-LTE. The full potential of MIMO systems can, however, only be achieved by means iterative MIMO decoding relying on soft-input soft-output (SISO) data detection. In this paper, we describe the first ASIC implementation of a SISO detector for iterative MIMO decoding. To this end, we propose a low-complexity minimum mean-squared error (MMSE) based parallel interference cancellation algorithm, develop a suitable VLSI architecture, and present a corresponding four-stream 1.5 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> detector chip in 90 nm CMOS technology. The fabricated ASIC includes all necessary preprocessing circuitry and exceeds the 600 Mb/s peak data-rate of IEEE 802.11n. A comparison with state-of-the-art MIMO-detector implementations demonstrates the performance benefits of our ASIC prototype in practical system-scenarios.

A Low Complexity MMSE Equalizer for OFDM Systems over Time-Varying Channels

A low complexity minimum mean squared error (MMSE) equalizer for orthogonal frequency division multiplexing (OFDM) systems over time-varying channels is presented. It uses a small matrix of dominant partial channel information and recursive calculation of matrix inverse to significantly reduce the complexity. Theoretical analysis and simulations results are provided to validate its significant performance or complexity advantages over the previously published MMSE equalizers.

Diversity Combining Options and Low-Complexity MMSE Equalization for Spread Spectrum OFDM Systems in Frequency Selective Fading Channels

This paper compares diversity combining schemes for the downlink of spread spectrum orthogonal frequency division multiplexing (SS-OFDM) systems in frequency selective fading channels. In particular, symbol-level combining after despreading is compared to chip-level combining under maximal ratio combining (MRC) of signals from different diversity branches and minimum mean-square error (MMSE) equalization of spreading sequences. Symbol-level combining takes place after the operations of MMSE equalization and despreading, whereas the operations of equalization and despreading occur after MRC if chip-level combining is used. Chip-level combining combines diversity samples in an efficient manner while reducing inter-code interference (self-interference) that results from the loss of orthogonality of spreading sequences due to a frequency selective channel. This method is shown to be superior to symbol-level combining when the diversity branches are uncorrelated, and when the branches differ only due to subcarrier interleaving. An MMSE equalization method with significantly reduced complexity for partially loaded systems is also presented, based on the premise of chip-level combining. Novel expressions for the bit error rate (BER) of the two methods, as well as the extension of the analysis to partially loaded systems are given. The extensions of chip-level combining and low-complexity equalization of a partially loaded system to an OFDM system with 2-dimensional spreading are also presented. The results are relevant to antenna diversity as well as temporal diversity achieved though re-transmission within an ARQ scheme.

Low-complexity MMSE turbo equalization: a possible solution for EDGE

This paper deals with a low complexity receiver scheme where equalization and channel decoding are jointly optimized in an iterative process. We derive the theoretical transfer function of the infinite length linear minimum mean square error (MMSE) equalizer with a priori information. A practical implementation is exposed which employs the fast Fourier transform (FFT) to compute the equalizer coefficients, resulting in a low-complexity receiver structure. The performance of the proposed scheme is investigated for the enhanced general packet radio service (EGPRS) radio link. Simulation results show that significant power gains may be achieved with only a few (3-4) iterations. These results demonstrate that MMSE turbo equalization is an attractive candidate for single-carrier broadband wireless transmissions in long delay-spread environments.

Bounding performance and suppressing intercarrier interference in wireless mobile OFDM

While rapid variations of the fading channel cause intercarrier interference (ICI) in orthogonal frequency-division multiplexing (OFDM), thereby degrading its performance considerably, they also introduce temporal diversity, which can be exploited to improve performance. We first derive a matched-filter bound (MFB) for OFDM transmissions over doubly selective Rayleigh fading channels, which benchmarks the best possible performance if ICI is completely canceled without noise enhancement. We then derive universal performance bounds which show that the time-varying channel causes most of the symbol energy to be distributed over a few subcarriers, and that the ICI power on a subcarrier mainly comes from several neighboring subcarriers. Based on this fact, we develop low-complexity minimum mean-square error (MMSE) and decision-feedback equalizer (DFE) receivers for ICI suppression. Simulations show that the DFE receiver can collect significant gains of ICI-impaired OFDM with affordable complexity. In the relatively low Doppler frequency region, the bit-error rate of the DFE receiver is close to the MFB.

Analysis of low-complexity windowed DFT-based MMSE channel estimator for OFDM systems

Low-complexity windowed discrete Fourier transform (DFT)-based minimum mean square error (MMSE) channel estimators are proposed and analyzed for both the interpolation and noninterpolation cases for orthogonal frequency-division multiplexing (OFDM) mobile communications systems. In the proposed method, the frequency domain data windowing is used to reduce the aliasing errors for the interpolation case and get better noise filtering performance for the noninterpolation case. The time domain MMSE weighting is also used to suppress the channel noise for both cases. Moreover, the optimal generalized Hanning window shape is searched to minimize the channel estimation mean square error (MSE). Analysis and simulation results show that the proposed method performance is close to the optimal MMSE estimator and is much better than the direct DFT-based estimator for both cases. Compared with the optimal MMSE estimator, however, the computation load of the proposed method can be significantly reduced because the IDFT/DFT transforms can be implemented with the fast algorithms IFFT/FFT.